Bayes's Theorem

Statement of Bayes's Theorem

Bayes's theorem is a statement on probability. For any two events A and B, the Bayes's theorem is given by

Here, P(A|B) is the conditional probability of A given B, and P(B|A) is the conditional probability of B given A. P(A)and P(B) are the marginal probabilities [Page 74]of events A and B, respectively. To see this theorem expressing how a probability is updated in the presence of new evidence, P(A) is considered as the prior probability of A, that is, the probability of A without taking any other information account. Then given the information on B, the updated probability of A, P(A|B)is calculated. P(A|B)isalsocalled posterior probability because it takes information of B into account.

There is an alternative formulation of Bayes's theorem. Note that based on the rule of total probability,

where P(A)and P Aþ (are the probabilities of event A occurring and not occurring, respectively, and P(B|A) is the conditional probability of B given Anot occurring. Then by substituting P(B) with Equation 2 in Equation 1, Equation 1 could be rewritten as

The application of Bayes's theorem in diagnostics and screening tests is based on Equation 3.

The above formulations of Bayes's theorem are for discrete events. For continuous distributions, there is a version of Bayes's theorem based on probability densities:

In this formula, f (y) is the prior distribution of parameter y; and f (y|Xi) is the posterior distribution of y, updated from f (y) by incorporating information from x. Parameter y is often the interest of inference. This formulation for continuous variable is widely used in Bayesian analysis.

Bayes's Theorem and Screening Test

In screening tests, there are several important quantities, including predictive value positive (PV+), predictive value negative (PV−), sensitivity, and specificity, that are expressed in terms of probabilities. These quantities are defined as follows:

• PV + of a screening test is the probability that a person has a disease given that the test is positive.
• PV− of a screening test is the probability that a person does not have a disease given that the test is negative.
• Sensitivity of a screening test is the probability that the test is positive given the person has a disease.
• Specificity of a screening test is the probability that the test is negative given the person does not have a disease.

Let A = disease, B = test positive, A = no disease, and B = test negative, then predictive value positive = P(A|B), predictive value negative = P(A|B), sensitivity = P(B|A), and specificity = P(B|A). If the proportion of disease in the general population is P(A), the relationship between predictive value positive, predictive value negative, sensitivity, and specificity can be expressed by using Bayes's theorem. With appropriate substitution of terms in Equation 3, PV+ and PV− could be given

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